- g is symmetric since it's the metric, thus the second term is symmetric.
- The first term is the product of two derivatives, so is symmetric (i.e. [itex]\partial^0\phi \partial^1\phi \equiv \partial^1\phi\partial^0\phi[/itex], and similarly for other values of mu and nu.)

Sometimes one has a product of a symmetric tensor with another tensor which is not symmetric nor antisymmetric, then one can show that the antisymmetric part of the second is killed by the first, the same thing occurs for the antisymmetric case, this is why we need to antisymmetrise and symmetrise tensors: to see which part remains and which part is killed...